The features that govern the stability and persistence of species interaction networks, such as food webs, remain elusive, but recent work suggests that the distribution and strength of trophic links play an important role. Potential omnivory-stability relationships have been investigated and debated extensively, but we still have a relatively poor understanding of how levels of omnivory relate to the stability of diverse food webs. Here, we use an evolutionary assembly model to investigate how different trade-offs in resource use influence both food web structure and dynamic stability during the assembly process. We build on a previous model by allowing speciation along with the evolution of two traits: body size and feeding-niche width. Across a wide range of conditions, the level of omnivory in a food web is positively related to its dynamic instability (variability and species turnover). Parameter values favoring omnivory also allow a wider range of phenotypes to invade, often displacing existing species. This high species turnover leaves signatures in reconstructed phylogenies, with shorter branches connecting extant species in more omnivorous food webs. Our findings suggest that features of the environment may influence both trophic structure and dynamic stability, leading to emergent omnivory-stability relationships.